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Effective Field Theory for Polyelectrons

Analysis of polyelectron systems using effective field theory to reproduce true theory for practical purposes. Constructing EFT for systems like positronium, exploring bound state energies, and comparing EFT with true theory. Improvements with local corrections and perturbative matching for scattering amplitudes.

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Effective Field Theory for Polyelectrons

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  1. zzzzz Effective Field Theory Analysis of Polyelectrons - - Paul McGrath + LLWI 2009

  2. + - Motivation: Di-positronium + - Constructing an effective field theory (EFT) What? Why? - Positronium as an example + Making use of an EFT Looking for bound state energies Generalized eigenvalue problem Comparison: EFT vs. true theory Paul McGrath LLWI 2009

  3. An EFT can reproduce the true theory for all practical purposes. Ignore details of structure below characteristic length (above characteristic energy scale) V(r) r a Paul McGrath LLWI 2009

  4. An EFT can reproduce the true theory for all practical purposes. Ignore details of structure below characteristic length (above characteristic energy scale) V(r) r a Uncertainty principle l Paul McGrath LLWI 2009

  5. An EFT can reproduce the true theory for all practical purposes. Ignore details of structure below characteristic length (above characteristic energy scale) V(r) r a Uncertainty principle l Paul McGrath LLWI 2009

  6. An EFT can reproduce the true theory for all practical purposes. Ignore details of structure below characteristic length (above characteristic energy scale) V(r) Veff(r) r r a a Uncertainty principle l Paul McGrath LLWI 2009

  7. - + Positronium: Introduce a cutoff to remove the divergence a -1 V(r) = - r r Divergence Paul McGrath LLWI 2009

  8. - + Positronium: Introduce a cutoff to remove the divergence a -1 V(r) = - r r Divergence 1 L -f(Lr) a f(Lr) r Veff(r,L) = - r Finite Paul McGrath LLWI 2009

  9. Improve an EFT with local corrections. a f(Lr) Veff(r,L) = - r Paul McGrath LLWI 2009

  10. Improve an EFT with local corrections. a 2 f(Lr) D + d2 g(Lr) + d1 g(Lr) + … Veff(r,L) = - r + + + … = Vtrue(r) + O((p/L)n) Paul McGrath LLWI 2009

  11. Improve an EFT with local corrections. a 2 f(Lr) D + d2 g(Lr) + d1 g(Lr) + … Veff(r,L) = - r + + + … = Vtrue(r) + O((p/L)n) Next step is to fix constants d1, d2, d3, … Perturbative Matching of Scattering Amplitudes Paul McGrath LLWI 2009

  12. Now we have a potential composed of simple terms. a 2 f(Lr) D + d2 g(Lr) + d1 g(Lr) + … Veff(r,L) = - r Variational Principle. ^ ^ Eo < F H F = F Heff F + O((p/L)n) -b2r2 e.g. F ~ e Paul McGrath LLWI 2009

  13. More parameters will give you a better upper bound on the energy. - bi2 r2 Gaussian Basis Fi ~ e ^ Fi Heff Fj = Eo FiFj Paul McGrath LLWI 2009

  14. More parameters will give you a better upper bound on the energy. - bi2 r2 Gaussian Basis Fi ~ e ^ Fi Heff Fj = Eo FiFj S S Hij xj = l Wij xj j j Paul McGrath LLWI 2009

  15. More parameters will give you a better upper bound on the energy. - bi2 r2 Gaussian Basis Fi ~ e ^ Fi Heff Fj = Eo FiFj a EFT (no corrections) EFT with O()2 corrections Exact 1/n2 Numerical True Theory n L 1 1 2 1/4 3 1/9 Basis Size = 50, a = 0.01, L = 10.0 Hill, arXiv:hep-ph/0008002v1 Paul McGrath LLWI 2009

  16. More parameters will give you a better upper bound on the energy. - bi2 r2 Gaussian Basis Fi ~ e ^ Fi Heff Fj = Eo FiFj a EFT (no corrections) EFT with O()2 corrections Exact 1/n2 Numerical True Theory n L 1 1 0.999999001 2 1/4 0.249999875 3 1/9 0.111111074 Basis Size = 50, a = 0.01, L = 10.0 Hill, arXiv:hep-ph/0008002v1 Paul McGrath LLWI 2009

  17. More parameters will give you a better upper bound on the energy. - bi2 r2 Gaussian Basis Fi ~ e ^ Fi Heff Fj = Eo FiFj a EFT (no corrections) EFT with O()2 corrections Exact 1/n2 Numerical True Theory n L 1 1 0.999999001 1.00000000040 2 1/4 0.249999875 0.25000000002 3 1/9 0.111111074 0.11111111101 Basis Size = 50, a = 0.01, L = 10.0 Hill, arXiv:hep-ph/0008002v1 Paul McGrath LLWI 2009

  18. More parameters will give you a better upper bound on the energy. - bi2 r2 Gaussian Basis Fi ~ e ^ Fi Heff Fj = Eo FiFj a EFT (no corrections) EFT with O()2 corrections Exact 1/n2 Numerical True Theory n L 1 1 0.999999001 1.00000000040 0.99999999971 2 1/4 0.249999875 0.25000000002 0.24999999997 3 1/9 0.111111074 0.11111111101 0.11111111103 Basis Size = 50, a = 0.01, L = 10.0 Hill, arXiv:hep-ph/0008002v1 Paul McGrath LLWI 2009

  19. More parameters will give you a better upper bound on the energy. - bi2 r2 Gaussian Basis Fi ~ e ^ Fi Heff Fj = Eo FiFj a EFT (no corrections) EFT with O()2 corrections Exact 1/n2 Numerical True Theory n L 1 1 0.999999001 1.00000000040 0.99999999971 2 1/4 0.249999875 0.25000000002 0.24999999997 3 1/9 0.111111074 0.11111111101 0.11111111103 Basis Size = 50, a = 0.01, L = 10.0 EFT better for Ps-ion Paul McGrath LLWI 2009

  20. Summary • An effective field theory can reproduce the true theory for all practical purposes within its range of validity • Using an effective field theory one can achieve the same precision with less analytical effort • Using an effective field theory one can achieve more precision with less computational effort • Di-positronium next – energy levels not known to extremely high precision - + + - Paul McGrath LLWI 2009

  21. Summary • An effective field theory can reproduce the true theory for all practical purposes within its range of validity • Using an effective field theory one can achieve the same precision with less analytical effort • Using an effective field theory one can achieve more precision with less computational effort • Di-positronium next – energy levels not known to extremely high precision - + + - Thank you! Paul McGrath LLWI 2009

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